# Uncertainty and symmetry bounds for the quantum total detection probability

## TL;DR

This paper derives bounds on the total detection probability in quantum systems, revealing how interference and symmetry influence detection success, with implications for quantum search processes.

## Contribution

It introduces an uncertainty relation linking detection probability and energy fluctuations, and establishes symmetry-based bounds on detection success in quantum systems.

## Key findings

- Uncertainty relation connects detection probability with energy variance.
- Symmetry bounds limit detection probability based on state equivalence.
- Bounds are validated against exact solutions for small systems.

## Abstract

We investigate a generic discrete quantum system prepared in state $|\psi_\text{in}\rangle$, under repeated detection attempts aimed to find the particle in state $|d\rangle$, for example a quantum walker on a finite graph searching for a node. For the corresponding classical random walk, the total detection probability $P_\text{det}$ is unity. Due to destructive interference, one may find initial states $|\psi_\text{in}\rangle$ with $P_\text{det}<1$. We first obtain an uncertainty relation which yields insight on this deviation from classical behavior, showing the relation between $P_\text{det}$ and energy fluctuations: $ \Delta P \,\mathrm{Var}[\hat{H}]_d \ge | \langle d| [\hat{H}, \hat{D}] | \psi_\text{in} \rangle |^2$ where $\Delta P = P_\text{det} - |\langle\psi_\text{in}|d\rangle |^2$, and $\hat{D} = |d\rangle\langle d|$ is the measurement projector. Secondly, exploiting symmetry we show that $P_\text{det}\le 1/\nu$ where the integer $\nu$ is the number of states equivalent to the initial state. These bounds are compared with the exact solution for small systems, obtained from an analysis of the dark and bright subspaces, showing the usefulness of the approach. The upper bounds works well even in large systems, and we show how to tighten the lower bound in this case.